The Shear-Induced Diffusivity of a Dilute Suspension in Bounded Systems

In a recent paper, Zurita-Gotor & Blawzdziewicz, 2007 identified the role of wall reflection on fluid and particle trajectories for the flow of bounded dilute suspensions. In unbounded systems particle and fluid trajectories are either open or closed, but in both cases tracers and particles return to their original streamlines. Thus such binary interactions do not yield self- or gradient-diffusion. The presence of a no-slip boundary, however, leads to a class of open “flipping” trajectories which are capable of producing dispersion, and this mechanism has been demonstrated to be the dominant source of shear induced self-diffusivity measured by Zarraga & Leighton, 2002 and Beimfohr et. al.,1993. This was the case even though the experimental particle diameter to gap width ratios were 1:20 or smaller.

In this work, we explore this phenomenon further by examining the influence of wall induced trajectories on both shear-induced self- and gradient-diffusivities. The trajectories are determined using the stresslet reflection from a single wall to obtain an analytical expression for the fluid and particle velocity distributions (e.g., Blake & Chwang, 1974). This is integrated to obtain the resulting diffusivities using the method of da Cunha & Hinch, 1996. With two walls, fluid and particle trajectories are obtained using an infinite sequence of stresslet reflections.  The diffusivities calculated from binary interactions between a tracer and a suspension sphere scale linearly with concentration as expected, consistent with experimental observations in the dilute limit. The self-diffusivity arising from one wall is given analytically by D_s = , which is independent of the tracer size. For large tracers, it is shown that the tracers interact with multiple particles which results in a decrease in the self-diffusivity. For two walls, the self-diffusivity is given analytically and is consistent with both the calculations of Zurita-Gotor & Blawzdziewicz, 2007 and measurements of Zarraga & Leighton, 2002.

Of considerable interest is the gradient diffusivity, which controls the flux due to a gradient in concentration. Due to the symmetry of the interactions, the wall reflection contribution to this diffusivity in both gradient and vorticity directions for equal size tracers is identically zero.  For fluid tracers, however, the gradient diffusivity in the velocity gradient direction scales as , while that in the vorticity direction is again zero. Surprisingly, the symmetry of both fluid-particle and particle-particle interactions in the presence of a wall leads to the possibility of a flux in vorticity direction proportional to the second derivative . This leading order term has the potential to destabilize a concentration polarization layer in the dilute limit.

References

Beimfohr, S., Looby, T., Leighton, D. T., Measurement of the shear-induced coefficient of self-diffusion in dilute suspensions, in Proceedings of the DOE/NSF Workshop on Flow of Particles and Fluids, Ithaca, NY, 1993.

Blake, J. R., Chwang, A. T., Fundamental singularities of viscous flow. Part I: The image systems in the vicinity of a stationary no-slip boundary, J. Eng. Math., 8(1), 23-29, 1974.

da Cunha, F.R., Hinch, E.J., Shear-induced dispersion in a dilute suspension of rough spheres, J. Fluid. Mech., 309, 211-223, 1996.

Zarraga, I.E., Leighton, D.T., Measurement of an unexpectedly large shear-induced self-diffusivity in a dilute suspension of spheres, Phys. Fluids, 14(7), 2194-2201, 2002.

Zurita-Gotor, M., Blawzdziewicz, J., Wajnryb, E. Swapping Trajectories: a new wall-induced cross-streamline particle migration mechanism in a dilute suspension of spheres, J. Fluid. Mech. 592, 447-469, 2007.